Dispersionless bands, such as Landau levels, serve as a good starting point for obtaining interesting correlated states when interactions are added. With this motivation in mind, we study a variety of dispersionless ("flat") band structures that arise in tight-binding Hamiltonians defined on hexagonal and kagome lattices with staggered fluxes. The flat bands and their neighboring dispersing bands have several notable features: (a) Flat bands can be isolated from other bands by breaking time

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... rsal symmetry, allowing for an extensive degeneracy when these bands are partially filled; (b) An isolated flat band corresponds to a critical point between regimes where the band is electron-like or hole-like, with an anomalous Hall conductance that changes sign across the transition; (c) When the gap between a flat band and two neighboring bands closes, the system is described by a single spin-1 conical-like spectrum, extending to higher angular momentum the spin-1/2 Dirac-like spectra in topological insulators and graphene; and (d) some configurations of parameters admit two isolated parallel flat bands, raising the possibility of exotic "heavy excitons"; (e) We find that the Chern number of the flat bands, in all instances that we study here, is zero.